(a) The first term of an Arithmetic Progression (AP) is 8, the ratio of the 7th term to the 9th term is 5 : 8, find the common difference of the AP. (b) A trader bought 30 baskets of pawpaw and 100 baskets of mangoes for N2,450.00. She sold the pawpaw at a profit of 40% and the mangoes at a profit of 30%. If her profit on the entire transaction was N855.00, find the (i) cost price of a basket of pawpaw ; (ii) selling price of the 100 baskets of mangoes.
Explanation
(a) \(T_{1} = a = -8\) \(\frac{T_{7}}{T_{9}} = \frac{5}{8}\) \(\frac{a + 6d}{a + 8d} = \frac{5}{8} \implies \frac{-8 + 6d}{-8 + 8d} = \frac{5}{8}\) \(-8 + 6d = \frac{5}{8}(-8 + 8d) \implies -8 + 6d = -5 + 5d\) \(6d - 5d = -5 + 8\) \(d = 3\) (b) Let : the cost price \(CP_{1}\) of a basket of pawpaw be w; the cost price \(CP_{2}\) of a basket of mangoes be m. Then \(30w + 100m = N2,450 ..... (1)\) Also, let : The selling price of the baskets of pawpaw be \(SP_{1}\); the selling price of the baskets of mangoes be \(SP_{2}\). Then \(SP_{1} + SP_{2} = N(2,450 + 855) = N3,305 ...... (2)\) %age profit on the sale of pawpaw = \(\frac{SP_{1} - CP_{1}}{CP_{1}} \times 100%\) i.e. \(40% = \frac{SP_{1} - 30w}{30w} \times 100%\) \(\implies 40 \times 3w = (SP_{1} - 30w) \times 10\) \(12w = SP_{1} - 30w\) \(SP_{1} = 12w + 30w = 42w\) %age profit on the sale of mangoes = \(\frac{SP_{2} - CP_{2}}{CP_{2}} \times 100%\) i.e. \(30% = \frac{SP_{2} - 100m}{100m} \times 100%\) \(\implies 30m = SP_{2} - 100m\) \(SP_{2} = 30m + 100m = 130m\) But \(SP_{1} + SP_{2} = N3,305\) \(\implies 42w + 130m = 3305 .... (2)\) From (1), \(m = \frac{2450 - 30w}{100}\) \(m = \frac{245 - 3w}{10}\) Substitute \(m = \frac{245 - 3w}{10}\) in (3) \(42w + 130(\frac{245 - 3w}{10}) = 3305\) \(42w + 3,185 - 39w = 3305\) \(3w = 3305 - 3185 = 120\) \(w = N40.00\) (i) Hence, the cost price of a basket of pawpaw = N40.00. (ii) From (2), \(SP_{1} + SP_{2} = 3305\) \(42w + 130m = 3305 \implies 42 \times 40 + 130m = 3305\) \(1680 + 130m = 3305\) \(130m = SP_{2} = 3305 - 1680\) = \(N1,625.00\) Hence, the selling point of 100 baskets of mangoes = N1,625.00.